3.470 \(\int \frac{2 x+x^2}{(1+x)^2} \, dx\)

Optimal. Leaf size=9 \[ \frac{x^2}{x+1} \]

[Out]

x^2/(1 + x)

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Rubi [A]  time = 0.0059376, antiderivative size = 7, normalized size of antiderivative = 0.78, number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {683} \[ x+\frac{1}{x+1} \]

Antiderivative was successfully verified.

[In]

Int[(2*x + x^2)/(1 + x)^2,x]

[Out]

x + (1 + x)^(-1)

Rule 683

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d +
e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[2*c*d - b*e,
0] && IGtQ[p, 0] &&  !(EqQ[m, 3] && NeQ[p, 1])

Rubi steps

\begin{align*} \int \frac{2 x+x^2}{(1+x)^2} \, dx &=\int \left (1-\frac{1}{(1+x)^2}\right ) \, dx\\ &=x+\frac{1}{1+x}\\ \end{align*}

Mathematica [A]  time = 0.0028421, size = 7, normalized size = 0.78 \[ x+\frac{1}{x+1} \]

Antiderivative was successfully verified.

[In]

Integrate[(2*x + x^2)/(1 + x)^2,x]

[Out]

x + (1 + x)^(-1)

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Maple [A]  time = 0.003, size = 8, normalized size = 0.9 \begin{align*} x+ \left ( 1+x \right ) ^{-1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2+2*x)/(1+x)^2,x)

[Out]

x+1/(1+x)

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Maxima [A]  time = 1.1354, size = 9, normalized size = 1. \begin{align*} x + \frac{1}{x + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2+2*x)/(1+x)^2,x, algorithm="maxima")

[Out]

x + 1/(x + 1)

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Fricas [A]  time = 1.17235, size = 31, normalized size = 3.44 \begin{align*} \frac{x^{2} + x + 1}{x + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2+2*x)/(1+x)^2,x, algorithm="fricas")

[Out]

(x^2 + x + 1)/(x + 1)

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Sympy [A]  time = 0.069685, size = 5, normalized size = 0.56 \begin{align*} x + \frac{1}{x + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**2+2*x)/(1+x)**2,x)

[Out]

x + 1/(x + 1)

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Giac [A]  time = 1.1462, size = 11, normalized size = 1.22 \begin{align*} x + \frac{1}{x + 1} + 1 \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2+2*x)/(1+x)^2,x, algorithm="giac")

[Out]

x + 1/(x + 1) + 1